The question of the boundedness of the spectral projectors from \(L^{p^{\prime }} \rightarrow L^p\)  is equivalent to that of the boundedness of the Fourier extension operator from \(L^2 \rightarrow L^p\) . This is true on the hyperbolic space, considered here, as well as on the Euclidean space, where this question was first considered. Optimal bounds were obtained by Tomas [1]} except for the endpoint case, which is due to Stein.
These works of Tomas and Stein were the starting point of the Fourier restriction theory, which has flourished since, and proved to be related to a number of mathematical fields, from geometric combinatorics to number theory. We refer to the textbook by Demeter [2]} for an account of these developments.
